   Chapter 8.2, Problem 14E

Chapter
Section
Textbook Problem

Find the exact area of the surface obtained by rotating the curve about the x-axis.14. x = 1 + 2y2, 1 ≤ y ≤ 2

To determine

To find: The exact area of the surface obtained by rotating the curve about x-axis.

Explanation

Given information:

The equation of the curve is x=1+2y2,1y2 .

The curve is bounded between y=1 and y=2 .

Calculation:

Show the equation of the curve.

x=1+2y2 (1)

Calculate the area of the surface obtained by rotating the curve about x-axis using the relation:

S=cd2πy1+(dxdy)2dy (2)

Here, S is the area of the surface obtained by rotating the curve about x-axis and

cyd .

Differentiate both sides of Equation (1) with respect to y.

dxdy=ddy(1+2y2)=(0+4y)=4y

Substitute 4y for dxdy , 1 for c, and 2 for d in Equation (2).

S=122πy1+(4y)2dy=122πy1+16y2dy (3)

Consider the value of the function u=1+16y2 (4)

Calculate the upper limit of the function u using Equation (4).

Substitute 2 for y in Equation (4).

u=1+16×22=65

Calculate the lower limit of the function u using Equation (4).

Substitute 1 for y in Equation (4).

u=1+16×12=17

Differentiate both sides of the Equation (4) with respect to y

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